Sketch function from derivatives and limits
Company: Capital One
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
Consider a twice-differentiable function f: R → R with:
• lim_{x→−∞} f(x) = 0 and lim_{x→+∞} f(x) = 5.
• f'(x) > 0 on (−∞, −2) and (1, 3); f'(x) < 0 on (−2, 1) and (3, ∞); f'(x) = 0 only at x ∈ {−2, 1, 3}.
• f''(x) < 0 on (−∞, −1) and (−1, 2); f''(x) > 0 on (2, ∞); f'' is undefined at x = 2, but f is continuous everywhere.
Tasks:
(a) Sketch a qualitatively correct graph of f, marking all local extrema and any inflection points.
(b) Is it possible for f to have two local maxima and one local minimum consistent with the above? Justify rigorously.
(c) Determine where an inflection point must occur, if any, and explain the sign of the slope immediately before and after x = 2.
Overview: This question evaluates understanding of calculus concepts—derivatives, concavity, limits, and continuity—and the ability to interpret derivative sign patterns to infer monotonicity and locate extrema.
Read the full Capital One Data Scientist interview experience this question came from
